The dynamic spin structure factor S(k,ω) of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function of ω it is shown to exhibit a power-law singularity at the threshold frequency defined by the energy of a magnon at given k. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.